1.A particle moves along the $x$-axis with $x(t)=\left(4t^{3}-6t^{2}\right)$ m, with $t$ in seconds. Its acceleration at $t=1.0$ s is

1. zero.
2. $-12$ m/s$^2$.
3. $12$ m/s$^2$.
4. $24$ m/s$^2$.

2.For a particle moving along a line, the area between the acceleration against time graph and the time axis, over some interval, gives

1. the change in velocity over that interval.
2. the displacement over that interval.
3. the change in speed over that interval.
4. the average acceleration over that interval.

3.Two vectors each have magnitude $6.0$. One points $30^\circ$ above the $+x$-axis and the other $30^\circ$ below it. The magnitude of their sum is

1. $12.0$.
2. $6.0$.
3. $8.5$.
4. $10.4$.

4.A particle’s velocity against time is a single straight line with negative slope, crossing zero at $t=4.0$ s. Over the interval from $t=0$ to $t=8.0$ s, its displacement is

1. equal in magnitude to the total distance it traveled.
2. zero.
3. negative, and equal in magnitude to the distance it traveled.
4. impossible to determine without knowing the slope.

5.Frame $S’$ moves at constant velocity relative to frame $S$. The acceleration of a particle measured in $S’$, compared with its acceleration measured in $S$, is

1. the same.
2. larger by the relative speed of the frames.
3. smaller by the relative speed of the frames.
4. different, unless the particle is at rest.

6.A projectile is launched from level ground with speed $v_0$ at an angle $\theta$ above the horizontal. When it returns to its launch height, its speed is

1. zero.
2. $v_0\cos\theta$.
3. $v_0$.
4. $v_0\sin\theta$.

7.A thin rod of length $L$ lies along the $x$-axis from $0$ to $L$ with linear mass density $\lambda(x)=\lambda_0\left(1-\dfrac{x}{L}\right)$. Its center of mass is at

1. $x=\dfrac{L}{2}$.
2. $x=\dfrac{L}{3}$.
3. $x=\dfrac{2L}{3}$.
4. $x=\dfrac{L}{4}$.

8.A block of mass $m$ slides down a rough incline of angle $\theta$ at constant velocity. The magnitude of the friction force exerted on it is

1. $\mu_k mg$.
2. zero, since the block is not accelerating.
3. $mg$.
4. $mg\sin\theta$.

9.Block A is pulled along a frictionless floor by a horizontal string of non-negligible mass, which trails behind it to block B. The whole arrangement accelerates. The tension in the string

1. is greatest where the string meets block A.
2. is the same at every point along the string.
3. is greatest where the string meets block B.
4. is zero at both ends and greatest in the middle.

10.A $2.0$ kg object moves along the $x$-axis with $v(t)=\left(4t^{2}-2t\right)$ m/s, with $t$ in seconds. The magnitude of the net force exerted on it at $t=1.5$ s is

1. $6.0$ N.
2. $12$ N.
3. $20$ N.
4. $10$ N.

11.The gravitational field strength at the surface of a uniform solid sphere of radius $R$ is $g$. At a distance $\dfrac{R}{4}$ from its center, the field strength is

1. $16g$.
2. $\dfrac{g}{4}$.
3. $4g$.
4. $\dfrac{g}{16}$.

12.A block rests on an incline whose angle is slowly increased. At the angle $\theta$ at which the block is just about to slip,

1. $\tan\theta=\mu_k$.
2. $\sin\theta=\mu_s$.
3. $\cos\theta=\mu_s$.
4. $\tan\theta=\mu_s$.

13.Two identical ideal springs, each of force constant $k$, are joined end to end in series and a block is hung from the lower one. The effective force constant of the pair is

1. $\dfrac{k}{2}$.
2. $2k$.
3. $k$.
4. $\dfrac{k}{4}$.

14.An object is released from rest in a fluid that exerts a resistive force $\vec{F}_r=-b\vec{v}$. As it falls, the magnitude of its acceleration

1. stays constant at $g$.
2. starts at zero and grows toward $g$.
3. starts at $g$ and falls toward zero.
4. is zero throughout, since it reaches terminal speed at once.

15.A car rounds a frictionless curve of radius $r$, banked at angle $\theta$, at the one speed for which no friction is needed. That speed satisfies

1. $v^{2}=gr\sin\theta$.
2. $v^{2}=gr\tan\theta$.
3. $v^{2}=gr\cos\theta$.
4. $v^{2}=gr$.

16.A particle moves along a line with $v(t)=\left(6t-t^{2}\right)$ m/s, with $t$ in seconds and $t\ge 0$. It reverses direction at

1. $t=3.0$ s.
2. $t=0$ only.
3. it never reverses direction.
4. $t=6.0$ s.

17.A particle moves along a line with velocity $v(t)$. Its displacement between $t=0$ and $t=T$ is

1. $\displaystyle\int_0^T v\,dt$.
2. $\displaystyle\int_0^T |v|\,dt$.
3. $v(T)-v(0)$.
4. $\dfrac{dv}{dt}$ evaluated at $t=T$.

18.Ignoring air resistance, the path $y(x)$ of a projectile launched at an angle to the horizontal is

1. a straight line.
2. a circular arc.
3. a parabola.
4. a hyperbola.

19.A ball is released from rest inside a train that is moving along a straight track at constant velocity. As seen by a passenger on the train, the ball falls

1. straight down.
2. backward, toward the rear of the train.
3. forward, toward the front of the train.
4. along a parabola, curving toward the rear.

20.Car A travels due east at $20$ m/s and car B travels due north at $15$ m/s, both measured relative to the ground. The velocity of car A relative to car B is

1. $5.0$ m/s, directed east.
2. $35$ m/s, directed north of east.
3. $25$ m/s, directed north of east.
4. $25$ m/s, directed south of east.

21.For a system of particles acted on by external forces, the center of mass moves as though

1. it carried the mass of the heaviest particle in the system.
2. all of the system’s mass were concentrated there and every external force acted at that point.
3. no forces acted on it at all.
4. it were attached to the heaviest particle.

22.A uniform rope of total mass $m$, length $\ell$ and linear density $\lambda$ hangs from a ceiling with nothing attached to its lower end. The tension a distance $y$ above the lower end is

1. $mg$.
2. $mgy$.
3. zero.
4. $\lambda g y$.

23.A $2.0$ kg object starts from rest and has acceleration $a(t)=3t$ $\mathrm{m/s^2}$, with $t$ in seconds. Its velocity at $t=2.0$ s is

1. $6.0$ m/s.
2. $3.0$ m/s.
3. $12$ m/s.
4. $1.5$ m/s.

24.The magnitude of the gravitational field at the exact center of a uniform solid sphere is

1. infinite, because the distance is zero.
2. equal to its value at the surface.
3. zero.
4. half its value at the surface.

25.A block is held against a vertical wall by a horizontal force pressing it into the wall. The block does not slide. The magnitude of the friction force exerted on the block by the wall is

1. $\mu_s$ times the horizontal force.
2. equal to the weight of the block.
3. zero, because the block is not moving.
4. $\mu_k$ times the horizontal force.

26.Two identical ideal springs, each of force constant $k$, are mounted side by side so that both stretch by the same amount when a load is applied. The effective force constant of the pair is

1. $\dfrac{k}{2}$.
2. $k$.
3. $\dfrac{k}{4}$.
4. $2k$.

27.An object released from rest falls through a fluid that exerts a resistive force $\vec{F}_r=-b\vec{v}$. It reaches terminal speed when

1. the magnitude of the resistive force equals the magnitude of the weight.
2. its velocity falls to zero.
3. its acceleration reaches $g$.
4. the resistive force falls to zero.

28.An object travels in a circle at constant speed. The net force exerted on it is

1. tangent to the circle, in the direction of motion.
2. zero, since the speed is not changing.
3. directed toward the center of the circle.
4. directed away from the center of the circle.

29.A satellite moves in a circular orbit of radius $R$ about a body of mass $M$. Its orbital speed is

1. $\sqrt{GMR}$.
2. $\sqrt{\dfrac{GM}{R}}$.
3. $\dfrac{GM}{R}$.
4. $\dfrac{GM}{R^{2}}$.

30.An object moves in a straight line at constant velocity. Which statement must be true?

1. No forces are exerted on it.
2. Exactly one force is exerted on it.
3. Every force exerted on it is vertical.
4. The vector sum of the forces exerted on it is zero.

1.A particle moves along the $x$-axis with velocity $v(t)=4t-t^{2}$ m/s, with $t$ in seconds. Its displacement between $t=0$ and $t=4.0$ s is

1. $16$ m.
2. $21.3$ m.
3. zero.
4. $10.7$ m.

2.A particle starts from rest at the origin and has acceleration $a(t)=6t$ m/s$^{2}$, with $t$ in seconds. Its position at $t=2.0$ s is

1. $4.0$ m.
2. $8.0$ m.
3. $12$ m.
4. $24$ m.

3.Two vectors are $\vec{A}=2\hat{\imath}+3\hat{\jmath}$ and $\vec{B}=4\hat{\imath}-\hat{\jmath}$. The component of $\vec{A}$ along the direction of $\vec{B}$ is closest to

1. $1.21$.
2. $3.61$.
3. $5.00$.
4. $0.29$.

4.The position of a particle is $x(t)=t^{3}-6t^{2}+9t$ m, with $t$ in seconds. The particle is momentarily at rest at

1. $t=1.0$ s only.
2. $t=2.0$ s only.
3. $t=1.0$ s and $t=3.0$ s.
4. $t=0$ and $t=3.0$ s.

5.One ship travels due north at $8.0$ m/s and a second travels $30^\circ$ east of north at $10$ m/s, both measured relative to the water. The speed of the second ship relative to the first is closest to

1. $12.8$ m/s.
2. $18$ m/s.
3. $2.0$ m/s.
4. $5.0$ m/s.

6.A particle moves in a plane with $x=3t$ m and $y=4t-5t^{2}$ m, with $t$ in seconds. The magnitude of its velocity at $t=0.40$ s is

1. $3.0$ m/s.
2. $4.0$ m/s.
3. $5.0$ m/s.
4. zero.

7.A thin rod of length $L$ lies along the $x$-axis from $x=0$ to $x=L$ and has linear mass density $\lambda(x)=\lambda_0\left(1+\dfrac{x}{L}\right)$. Its center of mass is at

1. $\dfrac{L}{2}$.
2. $\dfrac{7L}{12}$.
3. $\dfrac{5L}{9}$.
4. $\dfrac{2L}{3}$.

8.A uniform chain of mass $m$ and length $L$ hangs at rest from a ceiling with nothing attached to its lower end. The tension in the chain at a distance $y$ below the ceiling is

1. $\dfrac{mgy}{L}$.
2. $\dfrac{mg(L-y)}{L}$.
3. $mg$.
4. $\dfrac{mgy}{L-y}$.

9.Blocks of mass $2.0$ kg and $4.0$ kg rest in contact on a frictionless level floor. A horizontal force of $12$ N is exerted on the $4.0$ kg block, pushing it toward the $2.0$ kg block. The magnitude of the force the $2.0$ kg block exerts on the $4.0$ kg block is

1. $6.0$ N.
2. $8.0$ N.
3. $12$ N.
4. $4.0$ N.

10.A $2.0$ kg object moves along a straight line under a net force $F(t)=6t$ N, with $t$ in seconds, starting from rest. Its speed at $t=2.0$ s is

1. $6.0$ m/s.
2. $12$ m/s.
3. $24$ m/s.
4. $3.0$ m/s.

11.A uniform solid sphere has mass $M$ and radius $R$. The magnitude of the gravitational field at a distance $R/2$ from its center, compared with the field at its surface, is

1. four times.
2. one quarter.
3. one half.
4. twice.

12.A small block sits on a horizontal turntable at a distance of $0.20$ m from the axis. The coefficient of static friction between the block and the turntable is $0.30$. The greatest angular speed at which the block does not slide is closest to

1. $0.60$ rad/s.
2. $3.9$ rad/s.
3. $1.5$ rad/s.
4. $7.7$ rad/s.

13.A block of mass $m$ hangs at rest from an ideal spring of force constant $k$, which itself hangs from a second ideal spring of force constant $2k$ fixed to the ceiling. The total extension of the two springs together is

1. $\dfrac{mg}{3k}$.
2. $\dfrac{2mg}{3k}$.
3. $\dfrac{3mg}{k}$.
4. $\dfrac{3mg}{2k}$.

14.An object of mass $m$ is released from rest in a fluid that exerts a resistive force $\vec{F}_r=-b\vec{v}$. At the instant its speed is half its terminal speed, the magnitude of its acceleration is

1. $g/2$.
2. $g$.
3. zero.
4. $g/4$.

15.A ball of mass $m$ on a string of length $L$ swings in a vertical circle. At the lowest point of the circle it moves with speed $v$. The tension in the string there is

1. $mg$.
2. $\dfrac{mv^{2}}{L}$.
3. $m\left(g+\dfrac{v^{2}}{L}\right)$.
4. $m\left(\dfrac{v^{2}}{L}-g\right)$.

16.A particle moves along a line with velocity $v(t)=5-2t$ m/s, with $t$ in seconds. Its average velocity over the interval from $t=0$ to $t=5.0$ s is

1. $-5.0$ m/s.
2. zero.
3. $2.5$ m/s.
4. $5.0$ m/s.

17.A particle moves along the $x$-axis so that its velocity depends on position as $v=3x$, in SI units. Its acceleration at $x=2.0$ m is

1. $3.0$ m/s$^2$.
2. $6.0$ m/s$^2$.
3. $9.0$ m/s$^2$.
4. $18$ m/s$^2$.

18.A projectile is launched from level ground at $20$ m/s, $60^\circ$ above the horizontal. The instant at which its velocity is perpendicular to its launch velocity is closest to

1. $1.7$ s.
2. $2.3$ s.
3. $3.5$ s.
4. $1.0$ s.

19.A river of width $w$ flows at a steady speed $u$, and a boat can move at speed $v$ relative to the water, with $v$ larger than $u$. To reach the far bank in the shortest time, the boat should be steered

1. straight toward the opposite bank, whatever the current.
2. upstream, at the angle that cancels the current.
3. downstream, so that the current helps.
4. at $45^\circ$ to the bank.

20.The angle between $\vec{A}=3\hat{\imath}+4\hat{\jmath}$ and $\vec{B}=4\hat{\imath}+3\hat{\jmath}$ is closest to

1. $53^\circ$.
2. $0^\circ$.
3. $16^\circ$.
4. $37^\circ$.

21.Three particles lie in the $xy$-plane: $2.0$ kg at the origin, $3.0$ kg at $(4.0\ \mathrm{m},\,0)$ and $5.0$ kg at $(0,\,2.0\ \mathrm{m})$. The center of mass of the three is at

1. $(2.0\ \mathrm{m},\,1.0\ \mathrm{m})$.
2. $(1.2\ \mathrm{m},\,2.0\ \mathrm{m})$.
3. $(1.33\ \mathrm{m},\,0.67\ \mathrm{m})$.
4. $(1.2\ \mathrm{m},\,1.0\ \mathrm{m})$.

22.A block of mass $m$ is held motionless against a vertical wall by a horizontal force of magnitude $F$ pressing it into the wall. The magnitude of the normal force the wall exerts on the block is

1. $F$.
2. $mg$.
3. $\sqrt{F^{2}+m^{2}g^{2}}$.
4. zero.

23.A person of mass $m$ stands on the floor of an elevator that is accelerating upward with magnitude $a$. The magnitude of the force the person exerts on the floor is

1. $ma$.
2. $m(g-a)$.
3. $m(g+a)$.
4. $mg$.

24.A thin uniform spherical shell has mass $M$ and radius $R$. The magnitude of the gravitational field it produces at a distance $r$ from its center, for $r$ less than $R$, is

1. $\dfrac{GMr}{R^{3}}$.
2. zero.
3. $\dfrac{GM}{R^{2}}$.
4. $\dfrac{GM}{r^{2}}$.

25.A block of mass $m$ is pushed along a level floor at constant velocity by a force of magnitude $F$ directed at an angle $\theta$ above the horizontal. The coefficient of kinetic friction between the block and the floor is

1. $\tan\theta$.
2. $\dfrac{F\sin\theta}{mg}$.
3. $\dfrac{F\cos\theta}{mg}$.
4. $\dfrac{F\cos\theta}{mg-F\sin\theta}$.

26.An ideal spring of force constant $k$ is cut into two equal pieces. The force constant of each piece is

1. $2k$.
2. $\dfrac{k}{2}$.
3. $k$.
4. $4k$.

27.An object is thrown straight down with an initial speed greater than its terminal speed, in a medium exerting a resistive force $\vec{F}_r=-b\vec{v}$. As it falls, its speed

1. increases toward the terminal speed.
2. stays constant.
3. decreases toward the terminal speed.
4. decreases to zero.

28.A car rounds a curve of radius $r$ banked at angle $\theta$, at the one speed $v$ for which no friction is needed. The bank angle satisfies

1. $\sin\theta=\dfrac{v^{2}}{rg}$.
2. $\tan\theta=\dfrac{v^{2}}{rg}$.
3. $\cos\theta=\dfrac{v^{2}}{rg}$.
4. $\tan\theta=\dfrac{rg}{v^{2}}$.

29.A particle moves along a circle of radius $2.0$ m while its speed increases at a steady $3.0$ m/s$^2$. At the instant its speed is $4.0$ m/s, the magnitude of its total acceleration is

1. $11$ m/s$^2$.
2. $3.0$ m/s$^2$.
3. $8.0$ m/s$^2$.
4. $8.5$ m/s$^2$.

30.A particle moves with constant velocity. Which of the following must be true?

1. The vector sum of the forces exerted on it is zero.
2. No forces are exerted on it.
3. It is in a frame in which Newton’s laws do not hold.
4. The forces exerted on it are all equal in magnitude.